Playground
Interactive wave propagation: adjust frequency and wavelength sliders to see the wave move at v = fλ, with speed displayed in real time.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Wave speedoutput Phase speed of the wave in the medium | m/s | LT^-1 | 1 – 299792458 | |
| Frequency Number of wave cycles per second | Hz | T^-1 | 0.1 – 1000 | |
| Wavelength Distance between successive wave crests | m | L | 0.001 – 100 |
Deep dive
Derivation
By definition, wave speed is the distance a wave crest travels per unit time. In one period T, the crest moves exactly one wavelength λ. Therefore v = λ/T. Since frequency f = 1/T, substituting gives v = fλ. This is a kinematic identity — it holds for all periodic waves regardless of the medium or wave type.
Experimental verification
Verified by measuring the speed of sound using resonance tubes (Kundt, 1866), and for light using Fizeau's rotating cogwheel (1849) and Foucault's rotating mirror. Modern: laser interferometry and GPS timing confirm c = 299792458 m/s.
Common misconceptions
- Higher frequency means faster wave — frequency changes wavelength, not speed (in a given medium).
- Wave speed depends on amplitude — for linear waves, speed is independent of amplitude.
- v = fλ only applies to light — it applies to all periodic waves: sound, water, seismic, etc.
Real-world applications
- Sonar: knowing sound speed in water (≈1500 m/s) converts echo time to distance.
- Radio tuning: selecting a frequency determines the wavelength received by the antenna.
- Seismology: different wave speeds in rock layers reveal Earth's internal structure.
- Musical instruments: string length and tension set the wavelength and thus the pitch.
Worked examples
Lightning-to-thunder distance
Given:
- v:
- 343
- f:
- 100
Find: lambda
Solution
λ = v/f = 343/100 = 3.43 m
FM radio wavelength
Given:
- f:
- 100000000
- v:
- 299792458
Find: lambda
Solution
λ = c/f = 299792458 / 100000000 ≈ 3.0 m
Scenarios
What if…
- scenario:
- What if the medium changes?
- answer:
- Speed changes (e.g., sound: 343 m/s in air, 1500 m/s in water). Frequency stays the same, so wavelength adjusts: λ = v/f.
- scenario:
- What if frequency doubles?
- answer:
- At fixed wave speed, wavelength halves. The product fλ = v remains constant.
- scenario:
- What if v = c (speed of light)?
- answer:
- For electromagnetic waves in vacuum, fλ = 299792458 m/s. This is a universal constant — nothing with mass can reach it.
Limiting cases
- condition:
- f → ∞
- result:
- lambda → 0
- explanation:
- Infinite frequency means infinitesimally short wavelength at fixed speed.
- condition:
- lambda → ∞
- result:
- f → 0
- explanation:
- Infinitely long wavelength means near-zero frequency.
- condition:
- v = c (light in vacuum)
- result:
- f * lambda = 299792458
- explanation:
- For electromagnetic waves in vacuum, speed is fixed at c.
Context
Multiple contributors
A foundational relation recognized gradually through wave theory development by Huygens, Young, and Fresnel in the 17th-19th centuries.
Hook
You see lightning 3 seconds before hearing thunder — how far away was the strike?
Sound travels at 343 m/s in air. A 3-second delay means the lightning was about 1029 m away. Use v = f*lambda to relate wave speed to frequency and wavelength.
Dimensions:
- lhs:
- v → [LT⁻¹]
- rhs:
- f·λ → [T⁻¹]·[L] = [LT⁻¹]
- check:
- Both sides are [LT⁻¹] = m/s. ✓
Validity: Valid for all linear, non-dispersive waves. In dispersive media, phase velocity and group velocity differ; this equation gives phase velocity.